On a board with such knowledgable people, I'm surprised no one was able to get this one. However, I'm also honored to have stumped the band.
The answer is 1918. In 1918, the Brooklyn Robins began the season 0-9, while the New York Giants began theirs 9-0. On April 27, at Ebbets Field, Brooklyn beat New York, bringing their records to 1-9 and 9-1, respectively.
Along with 1962, it's the only other time in all of modern major sports history that a team with a record of 0-9 or worse beat a team with a record or 9-0 or better.

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